Math / Geometry

Question11) $A R C$ LenGTt $\widehat{A B}=$ 12) AREA OF SECTOR AOB =

Studdy Solution

STEP 1

1. The circle has a radius $R = 6 \, \text{cm}$ .
2. The central angle $\theta$ for the sector is $\frac{2\pi}{3}$ radians.

STEP 2

1. Calculate the arc length $\widehat{AB}$ .
2. Calculate the area of sector $AOB$ .

STEP 3

Recall the formula for the arc length of a circle:
$\text{Arc Length} = R \times \theta$
Substitute the given values:
$\text{Arc Length} = 6 \, \text{cm} \times \frac{2\pi}{3}$

STEP 4

Calculate the arc length:
$\text{Arc Length} = 6 \times \frac{2\pi}{3}$ $= 4\pi \, \text{cm}$

STEP 5

Recall the formula for the area of a sector:
$\text{Area of Sector} = \frac{1}{2} R^2 \theta$
Substitute the given values:
$\text{Area of Sector} = \frac{1}{2} \times 6^2 \times \frac{2\pi}{3}$

STEP 6

Calculate the area of the sector:
$\text{Area of Sector} = \frac{1}{2} \times 36 \times \frac{2\pi}{3}$ $= 12\pi \, \text{cm}^2$
The arc length $\widehat{AB}$ is:
$\boxed{4\pi \, \text{cm}}$
The area of sector $AOB$ is:
$\boxed{12\pi \, \text{cm}^2}$

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